Question
( 0 quad int frac{d x}{left(e^{x}+1right)left(2 e^{x}+3right)} )
0
( sqrt{frac{000000}{10000}} )
Let ( e^{x}=t Rightarrow x=log t )
diffenciating both sides:
( frac{d x}{d x}=frac{d log t}{d x} Rightarrow d x=frac{1}{t} d t )
fum 1 i: ( begin{aligned} int frac{d t}{t(t+1)(2 t+3)} & text { (setring by partial fraction) } end{aligned} )
wh ( frac{1}{t(t+1)(2 t+3)}=frac{A}{t}+frac{B}{t+1}+frac{c}{2 t+3} )
( =Aleft(2 t^{2}+5 t+3right)+Bleft(2 t^{2}+3 tright)+cleft(t^{2}+tright) )
On Conuparing like ferns both sides:
( 2 A+2 B+C=0 )
( S A+3 B+C=0 )
( 3 A=1 quad Rightarrow A=1 / 3 )
( frac{1}{t(1+t)(2 t+3)}=frac{1}{3 t}-frac{1}{(1+t)}+frac{4}{3(2 t+3)} )
Futegenting boll sides 2 ( int frac{d t}{t(1+t)(2 t+3)}=int frac{d t}{3 t}-int frac{d t}{1+t}+frac{4}{3} int frac{d t}{2 t+3} )
( =frac{1}{3} log t-log (1+t)+frac{2}{3} log (2 t+3) )
( =frac{1}{3} x-log left(1+e^{x}right)+frac{2}{3} log left(2 e^{x}+3right)+6 )

CRP-1819-JEEA-RP-PI-MA-4 dx is equal to The value of $10* + 1/2e* +3) 19 (A) x + in(e* + 1) – žin (2e* +3)+C (C) * -in (e* + 1) + In (20* + 3) +C - (B) x - In(e* + 1) + 3in (20* + (D) none of these IN
Solution
